src/HOL/Product_Type.thy
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(*  Title:      HOL/Product_Type.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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*)
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header {* Cartesian products *}
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theory Product_Type
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imports Typedef Inductive Fun
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keywords "inductive_set" "coinductive_set" :: thy_decl
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begin
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subsection {* @{typ bool} is a datatype *}
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rep_datatype True False by (auto intro: bool_induct)
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declare case_split [cases type: bool]
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  -- "prefer plain propositional version"
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lemma
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  shows [code]: "HOL.equal False P \<longleftrightarrow> \<not> P"
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    and [code]: "HOL.equal True P \<longleftrightarrow> P" 
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    and [code]: "HOL.equal P False \<longleftrightarrow> \<not> P"
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    and [code]: "HOL.equal P True \<longleftrightarrow> P"
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    and [code nbe]: "HOL.equal P P \<longleftrightarrow> True"
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  by (simp_all add: equal)
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lemma If_case_cert:
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  assumes "CASE \<equiv> (\<lambda>b. If b f g)"
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  shows "(CASE True \<equiv> f) &&& (CASE False \<equiv> g)"
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  using assms by simp_all
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setup {*
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  Code.add_case @{thm If_case_cert}
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*}
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code_const "HOL.equal \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool"
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  (Haskell infix 4 "==")
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code_instance bool :: equal
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  (Haskell -)
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subsection {* The @{text unit} type *}
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typedef (open) unit = "{True}"
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  by auto
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definition Unity :: unit  ("'(')")
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  where "() = Abs_unit True"
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lemma unit_eq [no_atp]: "u = ()"
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  by (induct u) (simp add: Unity_def)
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text {*
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  Simplification procedure for @{thm [source] unit_eq}.  Cannot use
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  this rule directly --- it loops!
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*}
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simproc_setup unit_eq ("x::unit") = {*
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  fn _ => fn _ => fn ct =>
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    if HOLogic.is_unit (term_of ct) then NONE
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    else SOME (mk_meta_eq @{thm unit_eq})
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*}
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rep_datatype "()" by simp
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lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
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  by simp
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lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
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  by (rule triv_forall_equality)
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text {*
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  This rewrite counters the effect of simproc @{text unit_eq} on @{term
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  [source] "%u::unit. f u"}, replacing it by @{term [source]
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  f} rather than by @{term [source] "%u. f ()"}.
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*}
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lemma unit_abs_eta_conv [simp, no_atp]: "(%u::unit. f ()) = f"
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  by (rule ext) simp
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lemma UNIV_unit [no_atp]:
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  "UNIV = {()}" by auto
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instantiation unit :: default
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begin
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definition "default = ()"
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instance ..
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end
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lemma [code]:
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  "HOL.equal (u\<Colon>unit) v \<longleftrightarrow> True" unfolding equal unit_eq [of u] unit_eq [of v] by rule+
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code_type unit
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  (SML "unit")
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  (OCaml "unit")
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  (Haskell "()")
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  (Scala "Unit")
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code_const Unity
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  (SML "()")
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  (OCaml "()")
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  (Haskell "()")
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  (Scala "()")
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code_instance unit :: equal
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  (Haskell -)
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code_const "HOL.equal \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"
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  (Haskell infix 4 "==")
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code_reserved SML
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  unit
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code_reserved OCaml
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  unit
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code_reserved Scala
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  Unit
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subsection {* The product type *}
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subsubsection {* Type definition *}
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definition Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" where
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  "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
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definition "prod = {f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}"
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typedef (open) ('a, 'b) prod (infixr "*" 20) = "prod :: ('a \<Rightarrow> 'b \<Rightarrow> bool) set"
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  unfolding prod_def by auto
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type_notation (xsymbols)
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  "prod"  ("(_ \<times>/ _)" [21, 20] 20)
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type_notation (HTML output)
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  "prod"  ("(_ \<times>/ _)" [21, 20] 20)
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definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" where
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  "Pair a b = Abs_prod (Pair_Rep a b)"
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rep_datatype Pair proof -
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  fix P :: "'a \<times> 'b \<Rightarrow> bool" and p
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  assume "\<And>a b. P (Pair a b)"
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  then show "P p" by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def)
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next
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  fix a c :: 'a and b d :: 'b
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  have "Pair_Rep a b = Pair_Rep c d \<longleftrightarrow> a = c \<and> b = d"
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    by (auto simp add: Pair_Rep_def fun_eq_iff)
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  moreover have "Pair_Rep a b \<in> prod" and "Pair_Rep c d \<in> prod"
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    by (auto simp add: prod_def)
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  ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d"
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    by (simp add: Pair_def Abs_prod_inject)
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qed
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declare prod.simps(2) [nitpick_simp del]
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declare prod.weak_case_cong [cong del]
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subsubsection {* Tuple syntax *}
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abbreviation (input) split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
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  "split \<equiv> prod_case"
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11777
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text {*
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  Patterns -- extends pre-defined type @{typ pttrn} used in
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  abstractions.
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*}
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nonterminal tuple_args and patterns
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syntax
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  "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
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  "_tuple_arg"  :: "'a => tuple_args"                   ("_")
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  "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
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  "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
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  ""            :: "pttrn => patterns"                  ("_")
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  "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
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translations
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  "(x, y)" == "CONST Pair x y"
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  "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
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  "%(x, y, zs). b" == "CONST prod_case (%x (y, zs). b)"
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  "%(x, y). b" == "CONST prod_case (%x y. b)"
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  "_abs (CONST Pair x y) t" => "%(x, y). t"
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  -- {* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
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     The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *}
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(*reconstruct pattern from (nested) splits, avoiding eta-contraction of body;
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  works best with enclosing "let", if "let" does not avoid eta-contraction*)
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print_translation {*
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let
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  fun split_tr' [Abs (x, T, t as (Abs abs))] =
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        (* split (%x y. t) => %(x,y) t *)
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        let
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          val (y, t') = Syntax_Trans.atomic_abs_tr' abs;
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          val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t');
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        in
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          Syntax.const @{syntax_const "_abs"} $
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            (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
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        end
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    | split_tr' [Abs (x, T, (s as Const (@{const_syntax prod_case}, _) $ t))] =
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        (* split (%x. (split (%y z. t))) => %(x,y,z). t *)
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        let
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          val Const (@{syntax_const "_abs"}, _) $
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            (Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t];
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          val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t');
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        in
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          Syntax.const @{syntax_const "_abs"} $
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            (Syntax.const @{syntax_const "_pattern"} $ x' $
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              (Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t''
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        end
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    | split_tr' [Const (@{const_syntax prod_case}, _) $ t] =
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        (* split (split (%x y z. t)) => %((x, y), z). t *)
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        split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)
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    | split_tr' [Const (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] =
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        (* split (%pttrn z. t) => %(pttrn,z). t *)
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        let val (z, t) = Syntax_Trans.atomic_abs_tr' abs in
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          Syntax.const @{syntax_const "_abs"} $
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            (Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t
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        end
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    | split_tr' _ = raise Match;
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in [(@{const_syntax prod_case}, split_tr')] end
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*}
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(* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
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typed_print_translation {*
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let
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  fun split_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match
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    | split_guess_names_tr' T [Abs (x, xT, t)] =
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        (case (head_of t) of
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          Const (@{const_syntax prod_case}, _) => raise Match
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        | _ =>
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          let 
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            val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
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            val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0);
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            val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, xT, t');
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          in
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            Syntax.const @{syntax_const "_abs"} $
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              (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
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          end)
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    | split_guess_names_tr' T [t] =
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        (case head_of t of
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          Const (@{const_syntax prod_case}, _) => raise Match
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        | _ =>
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          let
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            val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
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            val (y, t') =
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              Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0);
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            val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t');
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          in
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            Syntax.const @{syntax_const "_abs"} $
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              (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
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          end)
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    | split_guess_names_tr' _ _ = raise Match;
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in [(@{const_syntax prod_case}, split_guess_names_tr')] end
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*}
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(* Force eta-contraction for terms of the form "Q A (%p. prod_case P p)"
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   where Q is some bounded quantifier or set operator.
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   Reason: the above prints as "Q p : A. case p of (x,y) \<Rightarrow> P x y"
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   whereas we want "Q (x,y):A. P x y".
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   Otherwise prevent eta-contraction.
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*)
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print_translation {*
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let
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  fun contract Q f ts =
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    case ts of
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      [A, Abs(_, _, (s as Const (@{const_syntax prod_case},_) $ t) $ Bound 0)]
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      => if Term.is_dependent t then f ts else Syntax.const Q $ A $ s
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    | _ => f ts;
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  fun contract2 (Q,f) = (Q, contract Q f);
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  val pairs =
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    [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Ball} @{syntax_const "_Ball"},
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     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Bex} @{syntax_const "_Bex"},
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     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
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     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
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in map contract2 pairs end
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*}
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subsubsection {* Code generator setup *}
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code_type prod
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  (SML infix 2 "*")
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  (OCaml infix 2 "*")
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  (Haskell "!((_),/ (_))")
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  (Scala "((_),/ (_))")
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code_const Pair
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  (SML "!((_),/ (_))")
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  (OCaml "!((_),/ (_))")
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  (Haskell "!((_),/ (_))")
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  (Scala "!((_),/ (_))")
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code_instance prod :: equal
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  (Haskell -)
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code_const "HOL.equal \<Colon> 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
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  (Haskell infix 4 "==")
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subsubsection {* Fundamental operations and properties *}
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lemma surj_pair [simp]: "EX x y. p = (x, y)"
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  by (cases p) simp
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definition fst :: "'a \<times> 'b \<Rightarrow> 'a" where
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  "fst p = (case p of (a, b) \<Rightarrow> a)"
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definition snd :: "'a \<times> 'b \<Rightarrow> 'b" where
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  "snd p = (case p of (a, b) \<Rightarrow> b)"
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lemma fst_conv [simp, code]: "fst (a, b) = a"
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  unfolding fst_def by simp
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lemma snd_conv [simp, code]: "snd (a, b) = b"
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  unfolding snd_def by simp
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code_const fst and snd
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  (Haskell "fst" and "snd")
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   326
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lemma prod_case_unfold [nitpick_unfold]: "prod_case = (%c p. c (fst p) (snd p))"
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  by (simp add: fun_eq_iff split: prod.split)
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   329
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lemma fst_eqD: "fst (x, y) = a ==> x = a"
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  by simp
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02d75712061d got rid of ML proof scripts for Product_Type;
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lemma snd_eqD: "snd (x, y) = a ==> y = a"
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  by simp
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   335
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lemma pair_collapse [simp]: "(fst p, snd p) = p"
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  by (cases p) simp
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   338
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   339
lemmas surjective_pairing = pair_collapse [symmetric]
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   340
44066
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   341
lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
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  by (cases s, cases t) simp
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   343
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   344
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
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  by (simp add: prod_eq_iff)
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   346
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   347
lemma split_conv [simp, code]: "split f (a, b) = f a b"
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  by (fact prod.cases)
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   349
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   350
lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
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   351
  by (rule split_conv [THEN iffD2])
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   352
e8400e31528a more coherent theory structure; tuned headings
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   353
lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
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   354
  by (rule split_conv [THEN iffD1])
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   355
e8400e31528a more coherent theory structure; tuned headings
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   356
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
39302
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   357
  by (simp add: fun_eq_iff split: prod.split)
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   358
e8400e31528a more coherent theory structure; tuned headings
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   359
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
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   360
  -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
39302
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nipkow
parents: 39272
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   361
  by (simp add: fun_eq_iff split: prod.split)
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   362
e8400e31528a more coherent theory structure; tuned headings
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   363
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
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   364
  by (cases x) simp
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   365
e8400e31528a more coherent theory structure; tuned headings
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   366
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
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   367
  by (cases p) simp
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   368
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   369
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
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haftmann
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   370
  by (simp add: prod_case_unfold)
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   371
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   372
lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"
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   373
  -- {* Prevents simplification of @{term c}: much faster *}
40929
7ff03a5e044f theorem names generated by the (rep_)datatype command now have mandatory qualifiers
huffman
parents: 40702
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   374
  by (fact prod.weak_case_cong)
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   375
e8400e31528a more coherent theory structure; tuned headings
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   376
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
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   377
  by (simp add: split_eta)
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parents: 37136
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   378
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   379
lemma split_paired_all [no_atp]: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
11820
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
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   380
proof
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
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   381
  fix a b
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
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   382
  assume "!!x. PROP P x"
19535
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wenzelm
parents: 19179
diff changeset
   383
  then show "PROP P (a, b)" .
11820
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
parents: 11777
diff changeset
   384
next
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
parents: 11777
diff changeset
   385
  fix x
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
parents: 11777
diff changeset
   386
  assume "!!a b. PROP P (a, b)"
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19179
diff changeset
   387
  from `PROP P (fst x, snd x)` show "PROP P x" by simp
11820
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
parents: 11777
diff changeset
   388
qed
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
parents: 11777
diff changeset
   389
11838
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diff changeset
   390
text {*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
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   391
  The rule @{thm [source] split_paired_all} does not work with the
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   392
  Simplifier because it also affects premises in congrence rules,
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
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diff changeset
   393
  where this can lead to premises of the form @{text "!!a b. ... =
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
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diff changeset
   394
  ?P(a, b)"} which cannot be solved by reflexivity.
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wenzelm
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diff changeset
   395
*}
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   396
26358
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haftmann
parents: 26340
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   397
lemmas split_tupled_all = split_paired_all unit_all_eq2
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haftmann
parents: 26340
diff changeset
   398
26480
544cef16045b replaced 'ML_setup' by 'ML';
wenzelm
parents: 26358
diff changeset
   399
ML {*
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
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   400
  (* replace parameters of product type by individual component parameters *)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   401
  val safe_full_simp_tac = generic_simp_tac true (true, false, false);
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wenzelm
parents: 11820
diff changeset
   402
  local (* filtering with exists_paired_all is an essential optimization *)
16121
wenzelm
parents: 15570
diff changeset
   403
    fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   404
          can HOLogic.dest_prodT T orelse exists_paired_all t
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   405
      | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   406
      | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   407
      | exists_paired_all _ = false;
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   408
    val ss = HOL_basic_ss
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   409
      addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
43594
ef1ddc59b825 modernized some simproc setup;
wenzelm
parents: 42411
diff changeset
   410
      addsimprocs [@{simproc unit_eq}];
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   411
  in
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   412
    val split_all_tac = SUBGOAL (fn (t, i) =>
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   413
      if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   414
    val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   415
      if exists_paired_all t then full_simp_tac ss i else no_tac);
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   416
    fun split_all th =
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   417
   if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th;
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
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diff changeset
   418
  end;
26340
a85fe32e7b2f more antiquotations;
wenzelm
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diff changeset
   419
*}
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
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diff changeset
   420
26340
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wenzelm
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diff changeset
   421
declaration {* fn _ =>
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   422
  Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac))
16121
wenzelm
parents: 15570
diff changeset
   423
*}
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   424
47740
a8989fe9a3a5 added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents: 46950
diff changeset
   425
lemma split_paired_All [simp, no_atp]: "(ALL x. P x) = (ALL a b. P (a, b))"
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   426
  -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   427
  by fast
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   428
47740
a8989fe9a3a5 added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents: 46950
diff changeset
   429
lemma split_paired_Ex [simp, no_atp]: "(EX x. P x) = (EX a b. P (a, b))"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   430
  by fast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   431
47740
a8989fe9a3a5 added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents: 46950
diff changeset
   432
lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))"
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   433
  -- {* Can't be added to simpset: loops! *}
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   434
  by (simp add: split_eta)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   435
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   436
text {*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   437
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   438
  @{thm [source] split_eta} as a rewrite rule is not general enough,
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   439
  and using @{thm [source] cond_split_eta} directly would render some
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   440
  existing proofs very inefficient; similarly for @{text
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   441
  split_beta}.
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   442
*}
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   443
26480
544cef16045b replaced 'ML_setup' by 'ML';
wenzelm
parents: 26358
diff changeset
   444
ML {*
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   445
local
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   446
  val cond_split_eta_ss = HOL_basic_ss addsimps @{thms cond_split_eta};
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   447
  fun Pair_pat k 0 (Bound m) = (m = k)
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   448
    | Pair_pat k i (Const (@{const_name Pair},  _) $ Bound m $ t) =
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   449
        i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   450
    | Pair_pat _ _ _ = false;
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   451
  fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   452
    | no_args k i (t $ u) = no_args k i t andalso no_args k i u
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   453
    | no_args k i (Bound m) = m < k orelse m > k + i
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   454
    | no_args _ _ _ = true;
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   455
  fun split_pat tp i (Abs  (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   456
    | split_pat tp i (Const (@{const_name prod_case}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   457
    | split_pat tp i _ = NONE;
20044
92cc2f4c7335 simprocs: no theory argument -- use simpset context instead;
wenzelm
parents: 19656
diff changeset
   458
  fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   459
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18220
diff changeset
   460
        (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1)));
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   461
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   462
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   463
    | beta_term_pat k i (t $ u) =
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   464
        Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u)
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   465
    | beta_term_pat k i t = no_args k i t;
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   466
  fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   467
    | eta_term_pat _ _ _ = false;
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   468
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   469
    | subst arg k i (t $ u) =
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   470
        if Pair_pat k i (t $ u) then incr_boundvars k arg
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   471
        else (subst arg k i t $ subst arg k i u)
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   472
    | subst arg k i t = t;
43595
7ae4a23b5be6 modernized some simproc setup;
wenzelm
parents: 43594
diff changeset
   473
in
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   474
  fun beta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t) $ arg) =
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   475
        (case split_pat beta_term_pat 1 t of
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   476
          SOME (i, f) => SOME (metaeq ss s (subst arg 0 i f))
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15481
diff changeset
   477
        | NONE => NONE)
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   478
    | beta_proc _ _ = NONE;
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   479
  fun eta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t)) =
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   480
        (case split_pat eta_term_pat 1 t of
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   481
          SOME (_, ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end))
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15481
diff changeset
   482
        | NONE => NONE)
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   483
    | eta_proc _ _ = NONE;
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   484
end;
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   485
*}
43595
7ae4a23b5be6 modernized some simproc setup;
wenzelm
parents: 43594
diff changeset
   486
simproc_setup split_beta ("split f z") = {* fn _ => fn ss => fn ct => beta_proc ss (term_of ct) *}
7ae4a23b5be6 modernized some simproc setup;
wenzelm
parents: 43594
diff changeset
   487
simproc_setup split_eta ("split f") = {* fn _ => fn ss => fn ct => eta_proc ss (term_of ct) *}
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   488
26798
a9134a089106 split_beta is now declared as monotonicity rule, to allow bounded
berghofe
parents: 26588
diff changeset
   489
lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)"
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   490
  by (subst surjective_pairing, rule split_conv)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   491
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35427
diff changeset
   492
lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   493
  -- {* For use with @{text split} and the Simplifier. *}
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15422
diff changeset
   494
  by (insert surj_pair [of p], clarify, simp)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   495
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   496
text {*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   497
  @{thm [source] split_split} could be declared as @{text "[split]"}
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   498
  done after the Splitter has been speeded up significantly;
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   499
  precompute the constants involved and don't do anything unless the
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   500
  current goal contains one of those constants.
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   501
*}
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   502
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35427
diff changeset
   503
lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
14208
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   504
by (subst split_split, simp)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   505
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   506
text {*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   507
  \medskip @{term split} used as a logical connective or set former.
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   508
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   509
  \medskip These rules are for use with @{text blast}; could instead
40929
7ff03a5e044f theorem names generated by the (rep_)datatype command now have mandatory qualifiers
huffman
parents: 40702
diff changeset
   510
  call @{text simp} using @{thm [source] prod.split} as rewrite. *}
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   511
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   512
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   513
  apply (simp only: split_tupled_all)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   514
  apply (simp (no_asm_simp))
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   515
  done
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   516
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   517
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   518
  apply (simp only: split_tupled_all)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   519
  apply (simp (no_asm_simp))
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   520
  done
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   521
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   522
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   523
  by (induct p) auto
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   524
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   525
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   526
  by (induct p) auto
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   527
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   528
lemma splitE2:
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   529
  "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   530
proof -
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   531
  assume q: "Q (split P z)"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   532
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   533
  show R
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   534
    apply (rule r surjective_pairing)+
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   535
    apply (rule split_beta [THEN subst], rule q)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   536
    done
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   537
qed
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   538
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   539
lemma splitD': "split R (a,b) c ==> R a b c"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   540
  by simp
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   541
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   542
lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   543
  by simp
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   544
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   545
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
14208
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   546
by (simp only: split_tupled_all, simp)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   547
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   548
lemma mem_splitE:
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   549
  assumes major: "z \<in> split c p"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   550
    and cases: "\<And>x y. p = (x, y) \<Longrightarrow> z \<in> c x y \<Longrightarrow> Q"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   551
  shows Q
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   552
  by (rule major [unfolded prod_case_unfold] cases surjective_pairing)+
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   553
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   554
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   555
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   556
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   557
ML {*
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   558
local (* filtering with exists_p_split is an essential optimization *)
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   559
  fun exists_p_split (Const (@{const_name prod_case},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   560
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   561
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   562
    | exists_p_split _ = false;
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   563
  val ss = HOL_basic_ss addsimps @{thms split_conv};
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   564
in
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   565
val split_conv_tac = SUBGOAL (fn (t, i) =>
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   566
    if exists_p_split t then safe_full_simp_tac ss i else no_tac);
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   567
end;
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   568
*}
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   569
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   570
(* This prevents applications of splitE for already splitted arguments leading
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   571
   to quite time-consuming computations (in particular for nested tuples) *)
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   572
declaration {* fn _ =>
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   573
  Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac))
16121
wenzelm
parents: 15570
diff changeset
   574
*}
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   575
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35427
diff changeset
   576
lemma split_eta_SetCompr [simp,no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   577
  by (rule ext) fast
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   578
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35427
diff changeset
   579
lemma split_eta_SetCompr2 [simp,no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   580
  by (rule ext) fast
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   581
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   582
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   583
  -- {* Allows simplifications of nested splits in case of independent predicates. *}
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   584
  by (rule ext) blast
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   585
14337
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   586
(* Do NOT make this a simp rule as it
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   587
   a) only helps in special situations
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   588
   b) can lead to nontermination in the presence of split_def
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   589
*)
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   590
lemma split_comp_eq: 
20415
e3d2d7b01279 explicit type variables prevent empty sorts
paulson
parents: 20380
diff changeset
   591
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
e3d2d7b01279 explicit type variables prevent empty sorts
paulson
parents: 20380
diff changeset
   592
  shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   593
  by (rule ext) auto
14101
d25c23e46173 added upd_fst, upd_snd, some thms
oheimb
parents: 13480
diff changeset
   594
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   595
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   596
  apply (rule_tac x = "(a, b)" in image_eqI)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   597
   apply auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   598
  done
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   599
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   600
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   601
  by blast
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   602
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   603
(*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   604
the following  would be slightly more general,
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   605
but cannot be used as rewrite rule:
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   606
### Cannot add premise as rewrite rule because it contains (type) unknowns:
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   607
### ?y = .x
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   608
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
14208
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   609
by (rtac some_equality 1)
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   610
by ( Simp_tac 1)
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   611
by (split_all_tac 1)
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   612
by (Asm_full_simp_tac 1)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   613
qed "The_split_eq";
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   614
*)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   615
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   616
text {*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   617
  Setup of internal @{text split_rule}.
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   618
*}
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   619
45607
16b4f5774621 eliminated obsolete "standard";
wenzelm
parents: 45205
diff changeset
   620
lemmas prod_caseI = prod.cases [THEN iffD2]
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   621
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   622
lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   623
  by (fact splitI2)
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   624
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   625
lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   626
  by (fact splitI2')
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   627
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   628
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   629
  by (fact splitE)
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   630
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   631
lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   632
  by (fact splitE')
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   633
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   634
declare prod_caseI [intro!]
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   635
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 25885
diff changeset
   636
lemma prod_case_beta:
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 25885
diff changeset
   637
  "prod_case f p = f (fst p) (snd p)"
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   638
  by (fact split_beta)
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 25885
diff changeset
   639
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   640
lemma prod_cases3 [cases type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   641
  obtains (fields) a b c where "y = (a, b, c)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   642
  by (cases y, case_tac b) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   643
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   644
lemma prod_induct3 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   645
    "(!!a b c. P (a, b, c)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   646
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   647
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   648
lemma prod_cases4 [cases type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   649
  obtains (fields) a b c d where "y = (a, b, c, d)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   650
  by (cases y, case_tac c) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   651
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   652
lemma prod_induct4 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   653
    "(!!a b c d. P (a, b, c, d)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   654
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   655
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   656
lemma prod_cases5 [cases type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   657
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   658
  by (cases y, case_tac d) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   659
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   660
lemma prod_induct5 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   661
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   662
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   663
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   664
lemma prod_cases6 [cases type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   665
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   666
  by (cases y, case_tac e) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   667
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   668
lemma prod_induct6 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   669
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   670
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   671
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   672
lemma prod_cases7 [cases type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   673
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   674
  by (cases y, case_tac f) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   675
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   676
lemma prod_induct7 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   677
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   678
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   679
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   680
lemma split_def:
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   681
  "split = (\<lambda>c p. c (fst p) (snd p))"
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   682
  by (fact prod_case_unfold)
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   683
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   684
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   685
  "internal_split == split"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   686
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   687
lemma internal_split_conv: "internal_split c (a, b) = c a b"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   688
  by (simp only: internal_split_def split_conv)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   689
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47988
diff changeset
   690
ML_file "Tools/split_rule.ML"
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   691
setup Split_Rule.setup
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   692
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   693
hide_const internal_split
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   694
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   695
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   696
subsubsection {* Derived operations *}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   697
37387
3581483cca6c qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents: 37278
diff changeset
   698
definition curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where
3581483cca6c qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents: 37278
diff changeset
   699
  "curry = (\<lambda>c x y. c (x, y))"
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   700
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   701
lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   702
  by (simp add: curry_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   703
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   704
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   705
  by (simp add: curry_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   706
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   707
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   708
  by (simp add: curry_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   709
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   710
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   711
  by (simp add: curry_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   712
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   713
lemma curry_split [simp]: "curry (split f) = f"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   714
  by (simp add: curry_def split_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   715
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   716
lemma split_curry [simp]: "split (curry f) = f"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   717
  by (simp add: curry_def split_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   718
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   719
text {*
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   720
  The composition-uncurry combinator.
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   721
*}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   722
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   723
notation fcomp (infixl "\<circ>>" 60)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   724
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   725
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   726
  "f \<circ>\<rightarrow> g = (\<lambda>x. prod_case g (f x))"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   727
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   728
lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   729
  by (simp add: fun_eq_iff scomp_def prod_case_unfold)
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   730
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   731
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = prod_case g (f x)"
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   732
  by (simp add: scomp_unfold prod_case_unfold)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   733
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   734
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x"
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
   735
  by (simp add: fun_eq_iff)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   736
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   737
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x"
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
   738
  by (simp add: fun_eq_iff)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   739
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   740
lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   741
  by (simp add: fun_eq_iff scomp_unfold)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   742
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   743
lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   744
  by (simp add: fun_eq_iff scomp_unfold fcomp_def)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   745
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   746
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
   747
  by (simp add: fun_eq_iff scomp_unfold)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   748
31202
52d332f8f909 pretty printing of functional combinators for evaluation code
haftmann
parents: 30924
diff changeset
   749
code_const scomp
52d332f8f909 pretty printing of functional combinators for evaluation code
haftmann
parents: 30924
diff changeset
   750
  (Eval infixl 3 "#->")
52d332f8f909 pretty printing of functional combinators for evaluation code
haftmann
parents: 30924
diff changeset
   751
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   752
no_notation fcomp (infixl "\<circ>>" 60)
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   753
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   754
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   755
text {*
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   756
  @{term map_pair} --- action of the product functor upon
36664
6302f9ad7047 repaired comments where SOMEthing went utterly wrong (cf. 2b04504fcb69)
krauss
parents: 36622
diff changeset
   757
  functions.
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   758
*}
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   759
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   760
definition map_pair :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   761
  "map_pair f g = (\<lambda>(x, y). (f x, g y))"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   762
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   763
lemma map_pair_simp [simp, code]:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   764
  "map_pair f g (a, b) = (f a, g b)"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   765
  by (simp add: map_pair_def)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   766
41505
6d19301074cf "enriched_type" replaces less specific "type_lifting"
haftmann
parents: 41372
diff changeset
   767
enriched_type map_pair: map_pair
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
   768
  by (auto simp add: split_paired_all)
37278
307845cc7f51 added lemmas
nipkow
parents: 37166
diff changeset
   769
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   770
lemma fst_map_pair [simp]:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   771
  "fst (map_pair f g x) = f (fst x)"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   772
  by (cases x) simp_all
37278
307845cc7f51 added lemmas
nipkow
parents: 37166
diff changeset
   773
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   774
lemma snd_prod_fun [simp]:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   775
  "snd (map_pair f g x) = g (snd x)"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   776
  by (cases x) simp_all
37278
307845cc7f51 added lemmas
nipkow
parents: 37166
diff changeset
   777
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   778
lemma fst_comp_map_pair [simp]:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   779
  "fst \<circ> map_pair f g = f \<circ> fst"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   780
  by (rule ext) simp_all
37278
307845cc7f51 added lemmas
nipkow
parents: 37166
diff changeset
   781
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   782
lemma snd_comp_map_pair [simp]:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   783
  "snd \<circ> map_pair f g = g \<circ> snd"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   784
  by (rule ext) simp_all
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   785
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   786
lemma map_pair_compose:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   787
  "map_pair (f1 o f2) (g1 o g2) = (map_pair f1 g1 o map_pair f2 g2)"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   788
  by (rule ext) (simp add: map_pair.compositionality comp_def)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   789
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   790
lemma map_pair_ident [simp]:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   791
  "map_pair (%x. x) (%y. y) = (%z. z)"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   792
  by (rule ext) (simp add: map_pair.identity)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   793
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   794
lemma map_pair_imageI [intro]:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   795
  "(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_pair f g ` R"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   796
  by (rule image_eqI) simp_all
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   797
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   798
lemma prod_fun_imageE [elim!]:
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   799
  assumes major: "c \<in> map_pair f g ` R"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   800
    and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   801
  shows P
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   802
  apply (rule major [THEN imageE])
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   803
  apply (case_tac x)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   804
  apply (rule cases)
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   805
  apply simp_all
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   806
  done
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   807
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   808
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   809
  "apfst f = map_pair f id"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   810
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   811
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   812
  "apsnd f = map_pair id f"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   813
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   814
lemma apfst_conv [simp, code]:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   815
  "apfst f (x, y) = (f x, y)" 
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   816
  by (simp add: apfst_def)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   817
33638
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   818
lemma apsnd_conv [simp, code]:
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   819
  "apsnd f (x, y) = (x, f y)" 
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   820
  by (simp add: apsnd_def)
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   821
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   822
lemma fst_apfst [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   823
  "fst (apfst f x) = f (fst x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   824
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   825
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   826
lemma fst_apsnd [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   827
  "fst (apsnd f x) = fst x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   828
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   829
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   830
lemma snd_apfst [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   831
  "snd (apfst f x) = snd x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   832
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   833
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   834
lemma snd_apsnd [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   835
  "snd (apsnd f x) = f (snd x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   836
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   837
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   838
lemma apfst_compose:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   839
  "apfst f (apfst g x) = apfst (f \<circ> g) x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   840
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   841
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   842
lemma apsnd_compose:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   843
  "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   844
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   845
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   846
lemma apfst_apsnd [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   847
  "apfst f (apsnd g x) = (f (fst x), g (snd x))"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   848
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   849
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   850
lemma apsnd_apfst [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   851
  "apsnd f (apfst g x) = (g (fst x), f (snd x))"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   852
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   853
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   854
lemma apfst_id [simp] :
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   855
  "apfst id = id"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   856
  by (simp add: fun_eq_iff)
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   857
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   858
lemma apsnd_id [simp] :
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   859
  "apsnd id = id"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   860
  by (simp add: fun_eq_iff)
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   861
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   862
lemma apfst_eq_conv [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   863
  "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   864
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   865
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   866
lemma apsnd_eq_conv [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   867
  "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   868
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   869
33638
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   870
lemma apsnd_apfst_commute:
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   871
  "apsnd f (apfst g p) = apfst g (apsnd f p)"
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   872
  by simp
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   873
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   874
text {*
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   875
  Disjoint union of a family of sets -- Sigma.
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   876
*}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   877
45986
c9e50153e5ae moved various set operations to theory Set (resp. Product_Type)
haftmann
parents: 45696
diff changeset
   878
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   879
  Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   880
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   881
abbreviation
45986
c9e50153e5ae moved various set operations to theory Set (resp. Product_Type)
haftmann
parents: 45696
diff changeset
   882
  Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   883
    (infixr "<*>" 80) where
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   884
  "A <*> B == Sigma A (%_. B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   885
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   886
notation (xsymbols)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   887
  Times  (infixr "\<times>" 80)
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
   888
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   889
notation (HTML output)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   890
  Times  (infixr "\<times>" 80)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   891
45662
4f7c05990420 Hide Product_Type.Times - too precious an identifier
nipkow
parents: 45607
diff changeset
   892
hide_const (open) Times
4f7c05990420 Hide Product_Type.Times - too precious an identifier
nipkow
parents: 45607
diff changeset
   893
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   894
syntax
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34900
diff changeset
   895
  "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   896
translations
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34900
diff changeset
   897
  "SIGMA x:A. B" == "CONST Sigma A (%x. B)"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   898
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   899
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   900
  by (unfold Sigma_def) blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   901
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   902
lemma SigmaE [elim!]:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   903
    "[| c: Sigma A B;
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   904
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   905
     |] ==> P"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   906
  -- {* The general elimination rule. *}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   907
  by (unfold Sigma_def) blast
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
   908
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   909
text {*
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   910
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   911
  eigenvariables.
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   912
*}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   913
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   914
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   915
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   916
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   917
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   918
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   919
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   920
lemma SigmaE2:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   921
    "[| (a, b) : Sigma A B;
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   922
        [| a:A;  b:B(a) |] ==> P
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   923
     |] ==> P"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   924
  by blast
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
   925
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   926
lemma Sigma_cong:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   927
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   928
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   929
  by auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   930
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   931
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   932
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   933
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   934
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   935
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   936
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   937
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   938
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   939
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   940
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   941
  by auto
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
   942
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   943
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   944
  by auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   945
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   946
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   947
  by auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   948
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   949
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   950
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   951
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   952
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   953
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   954
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   955
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   956
  by (blast elim: equalityE)
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
   957
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   958
lemma SetCompr_Sigma_eq:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   959
    "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   960
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   961
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   962
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   963
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   964
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   965
lemma UN_Times_distrib:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   966
  "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   967
  -- {* Suggested by Pierre Chartier *}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   968
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   969
47740
a8989fe9a3a5 added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents: 46950
diff changeset
   970
lemma split_paired_Ball_Sigma [simp, no_atp]:
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   971
    "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   972
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   973
47740
a8989fe9a3a5 added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents: 46950
diff changeset
   974
lemma split_paired_Bex_Sigma [simp, no_atp]:
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   975
    "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   976
  by blast
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
   977
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   978
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   979
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   980
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   981
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   982
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   983
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   984
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   985
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   986
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   987
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   988
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   989
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   990
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   991
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   992
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   993
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   994
  by blast
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
   995
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   996
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   997
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   998
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   999
text {*
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1000
  Non-dependent versions are needed to avoid the need for higher-order
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1001
  matching, especially when the rules are re-oriented.
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1002
*}
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1003
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1004
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
28719
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1005
by blast
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1006
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1007
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
28719
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1008
by blast
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1009
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1010
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
28719
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1011
by blast
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1012
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1013
lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1014
  by auto
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1015
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1016
lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)"
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
  1017
  by force
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1018
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1019
lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)"
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
  1020
  by force
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1021
28719
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1022
lemma insert_times_insert[simp]:
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1023
  "insert a A \<times> insert b B =
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1024
   insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1025
by blast
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1026
33271
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 33089
diff changeset
  1027
lemma vimage_Times: "f -` (A \<times> B) = ((fst \<circ> f) -` A) \<inter> ((snd \<circ> f) -` B)"
47988
e4b69e10b990 tuned proofs;
wenzelm
parents: 47740
diff changeset
  1028
  apply auto
e4b69e10b990 tuned proofs;
wenzelm
parents: 47740
diff changeset
  1029
  apply (case_tac "f x")
e4b69e10b990 tuned proofs;
wenzelm
parents: 47740
diff changeset
  1030
  apply auto
e4b69e10b990 tuned proofs;
wenzelm
parents: 47740
diff changeset
  1031
  done
33271
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 33089
diff changeset
  1032
35822
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1033
lemma swap_inj_on:
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1034
  "inj_on (\<lambda>(i, j). (j, i)) A"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1035
  by (auto intro!: inj_onI)
35822
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1036
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1037
lemma swap_product:
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1038
  "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1039
  by (simp add: split_def image_def) blast
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1040
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1041
lemma image_split_eq_Sigma:
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1042
  "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
46128
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1043
proof (safe intro!: imageI)
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1044
  fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1045
  show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1046
    using * eq[symmetric] by auto
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1047
qed simp_all
35822
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1048
46128
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1049
definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1050
  [code_abbrev]: "product A B = A \<times> B"
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1051
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1052
hide_const (open) product
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1053
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1054
lemma member_product:
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1055
  "x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B"
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1056
  by (simp add: product_def)
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1057
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1058
text {* The following @{const map_pair} lemmas are due to Joachim Breitner: *}
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1059
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1060
lemma map_pair_inj_on:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1061
  assumes "inj_on f A" and "inj_on g B"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1062
  shows "inj_on (map_pair f g) (A \<times> B)"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1063
proof (rule inj_onI)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1064
  fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1065
  assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1066
  assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1067
  assume "map_pair f g x = map_pair f g y"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1068
  hence "fst (map_pair f g x) = fst (map_pair f g y)" by (auto)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1069
  hence "f (fst x) = f (fst y)" by (cases x,cases y,auto)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1070
  with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A`
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1071
  have "fst x = fst y" by (auto dest:dest:inj_onD)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1072
  moreover from `map_pair f g x = map_pair f g y`
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1073
  have "snd (map_pair f g x) = snd (map_pair f g y)" by (auto)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1074
  hence "g (snd x) = g (snd y)" by (cases x,cases y,auto)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1075
  with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B`
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1076
  have "snd x = snd y" by (auto dest:dest:inj_onD)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1077
  ultimately show "x = y" by(rule prod_eqI)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1078
qed
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1079
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1080
lemma map_pair_surj:
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40607
diff changeset
  1081
  fixes f :: "'a \<Rightarrow> 'b" and g :: "'c \<Rightarrow> 'd"
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1082
  assumes "surj f" and "surj g"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1083
  shows "surj (map_pair f g)"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1084
unfolding surj_def
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1085
proof
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1086
  fix y :: "'b \<times> 'd"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1087
  from `surj f` obtain a where "fst y = f a" by (auto elim:surjE)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1088
  moreover
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1089
  from `surj g` obtain b where "snd y = g b" by (auto elim:surjE)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1090
  ultimately have "(fst y, snd y) = map_pair f g (a,b)" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1091
  thus "\<exists>x. y = map_pair f g x" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1092
qed
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1093
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1094
lemma map_pair_surj_on:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1095
  assumes "f ` A = A'" and "g ` B = B'"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1096
  shows "map_pair f g ` (A \<times> B) = A' \<times> B'"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1097
unfolding image_def
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1098
proof(rule set_eqI,rule iffI)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1099
  fix x :: "'a \<times> 'c"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1100
  assume "x \<in> {y\<Colon>'a \<times> 'c. \<exists>x\<Colon>'b \<times> 'd\<in>A \<times> B. y = map_pair f g x}"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1101
  then obtain y where "y \<in> A \<times> B" and "x = map_pair f g y" by blast
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1102
  from `image f A = A'` and `y \<in> A \<times> B` have "f (fst y) \<in> A'" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1103
  moreover from `image g B = B'` and `y \<in> A \<times> B` have "g (snd y) \<in> B'" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1104
  ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1105
  with `x = map_pair f g y` show "x \<in> A' \<times> B'" by (cases y, auto)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1106
next
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1107
  fix x :: "'a \<times> 'c"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1108
  assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1109
  from `image f A = A'` and `fst x \<in> A'` have "fst x \<in> image f A" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1110
  then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1111
  moreover from `image g B = B'` and `snd x \<in> B'`
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1112
  obtain b where "b \<in> B" and "snd x = g b" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1113
  ultimately have "(fst x, snd x) = map_pair f g (a,b)" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1114
  moreover from `a \<in> A` and  `b \<in> B` have "(a , b) \<in> A \<times> B" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1115
  ultimately have "\<exists>y \<in> A \<times> B. x = map_pair f g y" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1116
  thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_pair f g y}" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1117
qed
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1118
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1119
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1120
subsection {* Inductively defined sets *}
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1121
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47988
diff changeset
  1122
ML_file "Tools/inductive_set.ML"
31723
f5cafe803b55 discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents: 31667
diff changeset
  1123
setup Inductive_Set.setup
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
  1124
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1125
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1126
subsection {* Legacy theorem bindings and duplicates *}
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1127
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1128
lemma PairE:
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1129
  obtains x y where "p = (x, y)"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1130
  by (fact prod.exhaust)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1131
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1132
lemma Pair_inject:
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1133
  assumes "(a, b) = (a', b')"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1134
    and "a = a' ==> b = b' ==> R"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1135
  shows R
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1136
  using assms by simp
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1137
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1138
lemmas Pair_eq = prod.inject
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1139
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1140
lemmas split = split_conv  -- {* for backwards compatibility *}
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1141
44066
d74182c93f04 rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents: 43866
diff changeset
  1142
lemmas Pair_fst_snd_eq = prod_eq_iff
d74182c93f04 rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents: 43866
diff changeset
  1143
45204
5e4a1270c000 hide typedef-generated constants Product_Type.prod and Sum_Type.sum
huffman
parents: 44921
diff changeset
  1144
hide_const (open) prod
5e4a1270c000 hide typedef-generated constants Product_Type.prod and Sum_Type.sum
huffman
parents: 44921
diff changeset
  1145
10213
01c2744a3786 *** empty log message ***
nipkow
parents:
diff changeset
  1146
end